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The Smaller (The Smaller (SALISALI ) and the ) and the Generalized (Generalized (GALIGALI ) Alignment ) Alignment

Index methods of chaos detectionIndex methods of chaos detection

Haris SkokosPhysics Department, Aristotle University of Thessaloniki

Thessaloniki, Greece

E-mail: [email protected]: http://users.auth.gr/hskokos/

Main contributors:Tassos Bountis, Chris Antonopoulos, Thanos Manos, Enrico Gerlach

OutlineOutline• Smaller ALignment Index – SALI

�Definition�Behavior for chaotic and regular motion�Applications

• Generalized ALignment Index – GALI�Definition - Relation to SALI�Behavior for chaotic and regular motion�Applications�Global dynamics�Motion on low-dimensional tori

• Summary

Definition of Smaller Definition of Smaller Alignment Index (SALI)Alignment Index (SALI)

Consider the 2N-dimensionalphase space of a conservative dynamical system(symplectic map or Hamiltonian flow).

An orbit in that space with initial condition :P(0)=(x1(0), x2(0),…,x2N(0))

and a deviation vectorv(0)=(δx1(0), δx2(0),…, δx2N(0))

The evolution in time (in maps the time is discrete and is equal to thenumber n of the iterations) of a deviation vectoris defined by:•the variational equations (for Hamiltonian flows) and•the equations of thetangent map(for mappings)

Definition of SALIDefinition of SALIWe follow the evolution in time of two different initialdeviation vectors (v1(0), v2(0)), and define SALI (Ch.S.2001, J. Phys. A) as:

When the two vectors becomecollinear

SALI(t) → 0

{ }ˆ ˆ ˆ ˆ1 2 1 2SALI(t) = min v (t) + v (t) , v (t) - v (t)

ˆ 11

1

v (t)v (t) =

v (t)

where

Behavior of SALI for Behavior of SALI for chaotic motionchaotic motionFor chaotic orbits the two initiallydifferent deviation vectors tend tocoincide with the direction definedby the maximum Lyapunovexponent.


P(0)

P(t)


ˆ 1v (0)

ˆ 2v (0)

P(0)

P(t)

2v (t)

1v (t)


ˆ 1v (0)

ˆ 2v (0)

P(0)

P(t)

2v (t)

1v (t)


ˆ 1v (0)

ˆ 2v (0)

ˆ 2v (t)

ˆ 1v (t)

P(0)

P(t)

2v (t)

1v (t)


ˆ 1v (0)

ˆ 2v (0)

ˆ 2v (t)

ˆ 1v (t)

P(0)

P(t)

SALI(0)

2v (t)

1v (t)


ˆ 1v (0)

ˆ 2v (0)

ˆ 2v (t)

ˆ 1v (t)

P(0)

P(t)

SALI(0)

SALI(t)

Behavior of SALI for Behavior of SALI for chaotic motionchaotic motionThe evolution of a deviation vector can be approximated by:

ˆ ˆ ˆ≈∑ i 1 2

nσ t σ t σ t(1) (1) (1)

1 i i 1 1 2 2i=1

v (t) = c e u c e u + c e u

In this approximation, we derive a leading order estimate of the ratio

ˆ ˆˆ ˆ≈ ± +

1 2

1 2

1

σ t σ t(1) (1) (1)-(σ -σ )t1 1 1 2 2 2

1 2σ t(1) (1)1 1 1

v (t) c e u + c e u c= u e u

v (t) c e cand an analogous expression for v2

ˆ ˆˆ ˆ≈ ± +

1 2

1 2

1

σ t σ t(2) (2) (2)-(σ -σ )t2 1 1 2 2 2

1 2σ t(2) (2)2 1 1

v (t) c e u + c e u c= u e u

v (t) c e cSo we get:

≈ ±

1 2

(1) (2)-(σ -σ )t1 2 1 2 2 2

(1) (2)1 2 1 2 1 1

v (t) v (t) v (t) v (t) c cSALI(t) = min + , - e

v (t) v (t) v (t) v (t) c c

where σ1>σ2≥… ≥ σn are the Lyapunov exponents, and j=1, 2, …, 2N the corresponding eigendirections.

ˆ ju

Behavior of SALI for Behavior of SALI for chaotic motionchaotic motion

∑3

2 2 2 2ii i 1 2 1 3

i=1

ωH = (q + p )+ q q + q q

2

We test the validity of the approximation SALIµe-(σ1-σ2)t (Ch.S.,Antonopoulos, Bountis, Vrahatis, 2004, J. Phys. A) for a chaotic orbitof the 3D Hamiltonian

with ω1=1, ω2=1.4142, ω3=1.7321, Η=0.09

σ1ª0.037

σ2ª0.011

slope=-(σ1-σ2)/ln(10)

Behavior of SALI for Behavior of SALI for regular motionregular motionRegular motion occurs on a torus and two different initialdeviation vectors become tangent to the torus, generallyhaving different directions.

P(0)


P(t)

P(0)


P(t)

P(0)


ˆ 2v (0)ˆ 1v (0)

P(t)

P(0)


ˆ 2v (0)ˆ 1v (0)

ˆ 1v (0)

ˆ 2v (0)

Applications Applications –– HénonHénon--Heiles systemHeiles system

For E=1/8 we consider the orbits with initial conditions:Regular orbit, x=0, y=0.55, px=0.2417, py=0Chaotic orbit, x=0, y=-0.016, px=0.49974, py=0Chaotic orbit, x=0, y=-0.01344, px=0.49982, py=0

As an example, we consider the 2D Hénon-Heiles system:

Applications Applications –– HénonHénon--Heiles systemHeiles system

y

py

Applications Applications –– 4D Accelerator map4D Accelerator mapWe consider the 4D symplectic map

′ ′ × ′ ′

11 1 12 2

2 1 32 1 1

33 2 2

4 1 34 2 2

xx cosω -sinω 0 0

x + x - xx sinω cosω 0 0 =

xx 0 0 cosω -sinω

x - 2x xx 0 0 sinω cosωdescribing the instantaneous sextupole ‘kicks’experienced by aparticle as itpasses through an accelerator(Turchetti & Scandale 1991, Bountis &Tompaidis 1991, Vrahatis et al. 1996, 1997).

x1 and x3 are the particle’s deflections from the ideal circular orbit, in thehorizontal and vertical directions respectively.x2 and x4 are the associated momenta

ω1, ω2 are related to the accelerator’s tunes qx, qy by ω1=2πqx, ω2=2πqy

Our goal is to estimate theregion of stability of the particle’s motion, the so-called dynamic aperture of the beam(Bountis, Ch.S., 2006, Nucl. Inst Meth.Phys Res. A)and to increase its size using chaos control techniques(Boreaux,Carletti, Ch.S., Vittot, 2012, Com. Nonlin. Sci. Num. Sim. –Boreaux, Carletti,Ch.S., Papaphilippou, Vittot, 2012, Int. J. Bif. Chaos).

4D Accelerator map 4D Accelerator map –– ""GlobalGlobal"" study study Regions ofdifferent values of the SALI on the subspacex2(0)=x4(0)=0, after 105 iterations (qx=0.61803 qy=0.4152)

4D map Controlled 4D map

log(SALI)

4D Accelerator map 4D Accelerator map –– ""GlobalGlobal"" study study Increase of the dynamic aperture

We evolve many orbits in 4D hyperspheres of radius rcentered at x1=x2=x3=x4=0, for 105 iterations.

4D mapControlled 4D map

Regular orbits

Chaotic orbits

Applications Applications –– 2D map2D map′

′1 1 2

2 2 1 2

x = x + x (mod 2π)

x = x - ν sin(x + x )

-3 -2 -1 0 1 2 3X

1

-3

-2

-1

0

1

2

3

X 2 A B

Forν=0.5 we consider the orbits:regular orbit A with initial conditions x1=2, x2=0.chaotic orbit B with initial conditions x1=3, x2=0.

Behavior of SALIBehavior of SALI2D maps

SALI→0 both for regular and chaotic orbitsfollowing, however, completely different time rates whichallows us to distinguish between the two cases.

Hamiltonian flows and multidimensional mapsSALI→0 for chaotic orbits

SALI→constant ≠ 0 for regular orbits

QuestionsQuestions

• Can rapidly reveal the nature of chaotic orbits withσ1ªσ2 (SALIµe-(σ1-σ2)t)?

• Depends on several Lyapunov exponents for chaoticorbits?

• Exhibits power-law decay for regular orbits dependingon the dimensionality of the tangent space of thereference orbit as for 2D maps?

Can we generalize SALI so that the new index:

Definition of Generalized Definition of Generalized Alignment Index (GALI)Alignment Index (GALI)

SALI effectively measures the ‘area’ of the parallelogramformed by the two deviation vectors.

ˆ 1v (0)

ˆ 2v (0)

ˆ 2v (t)

ˆ 1v (t)

P(0)

P(t)



ˆ 1v (0)

ˆ 2v (0)

ˆ 2v (t)

ˆ 1v (t)

P(0)

P(t)



{ }

ˆ ˆ ˆ ˆˆ ˆ

ˆ ˆ ˆ ˆ

⋅ += = =

+⋅ ⇒

∝

1 2 1 21 2

1 2 1 2

v - v v vArea v ^ v

2max v - v , v v

SALI2

Area SALI

Definition of GALIDefinition of GALIIn the case of an N degree of freedomHamiltonian systemora 2N symplectic map we followthe evolution of

k deviation vectors with 2≤k≤2N,

and define (Ch.S., Bountis, Antonopoulos, 2007, Physica D)the Generalized Alignment Index (GALI) of order k :

ˆ ˆ ˆ∧ ∧ ∧k 1 2 kGALI (t) = v (t) v (t) ... v (t)

ˆ 11

1

v (t)v (t) =

v (t)

where

Wedge productWedge productWe consider asa basis of the 2N-dimensional tangent space of thesystemthe usualset of orthonormal vectors:

ˆ ˆ ˆ1 2 2Ne = (1,0,0, ...,0), e = (0,1,0, ...,0), ..., e = (0,0,0, ...,1)Then for k deviation vectors we have:

ˆˆ

ˆˆ

ˆˆ

⋅

11 12 1 2N 11

21 22 2 2N 22

k1 k2 k 2N 2Nk

v v v ev

v v v ev =

v v v ev

⋯

⋯

⋮ ⋮ ⋮ ⋮⋮

⋯

ˆ ˆ ˆˆ ˆ ˆ≤ ≤

∧ ∧ ∧ ∧ ∧ ∧∑

1 2 k

1 2 k

1 2 k

1 2 k

1 2 k

1 i 1 i 1 i

2 i 2 i 2 i1 2 k i i i

1 i <i < <i 2N

k i k i k i

v v v

v v vv v v = e e e

v v v⋯

⋯

⋯⋯ ⋯

⋮ ⋮ ⋮

⋯

Norm of wedge productNorm of wedge product

We define as‘norm’ of the wedge product the quantity :

ˆ ˆ ˆ≤ ≤

∧ ∧ ∧

∑

1 2 k

1 2 k

1 2 k

1 2 k

1/22

1 i 1 i 1 i

2 i 2 i 2 i1 2 k

1 i <i < <i 2N

k i k i k i

v v v

v v vv v v =

v v v⋯

⋯

⋯⋯

⋮ ⋮ ⋮

⋯

Computation of Computation of GALI GALI -- ExampleExampleLet us compute GALI3 in the case of 2D Hamiltonian system(4-dimensional phase space).

ˆˆ

ˆˆ

ˆˆ

ˆ

= ⋅

11 11 12 13 14

22 21 22 23 24

33 31 32 33 34

4

ev v v v v

ev v v v v

ev v v v v

e


ˆˆ

ˆˆ

ˆˆ

ˆ

= ⋅

11 11 12 13 14

22 21 22 23 24

33 31 32 33 34

4

ev v v v v

ev v v v v

ev v v v v

e

ˆ ˆ ˆ

∧ ∧

2

11 12 13

3 1 2 3 21 22 23

31 32 33

v v v

GALI = v v v = v v v

v v v

++++

Columns 1 2 3


ˆˆ

ˆˆ

ˆˆ

ˆ

= ⋅

11 11 12 13 14

22 21 22 23 24

33 31 32 33 34

4

ev v v v v

ev v v v v

ev v v v v

e

ˆ ˆ ˆ

∧ ∧

2

11 12 13

3 1 2 3 21 22 23

31 32 33

v v v


v v v

++++

Columns 1 2 3 2

11 12 14

21 22 24

31 32 34

v v v

v v v +

v v v

1 2 4


ˆˆ

ˆˆ

ˆˆ

ˆ

= ⋅

11 11 12 13 14

22 21 22 23 24

33 31 32 33 34

4

ev v v v v

ev v v v v

ev v v v v

e

ˆ ˆ ˆ

∧ ∧

2

11 12 13

3 1 2 3 21 22 23

31 32 33

v v v


v v v

++++

Columns 1 2 3 2

11 12 14

21 22 24

31 32 34

v v v

v v v +

v v v

1 2 4

2

11 13 14

21 23 24

31 33 34

v v v

v v v +

v v v1 3 4


ˆˆ

ˆˆ

ˆˆ

ˆ

= ⋅

11 11 12 13 14

22 21 22 23 24

33 31 32 33 34

4

ev v v v v

ev v v v v

ev v v v v

e

ˆ ˆ ˆ

∧ ∧

2

11 12 13

3 1 2 3 21 22 23

31 32 33

v v v


v v v

++++

Columns 1 2 3 2

11 12 14

21 22 24

31 32 34

v v v

v v v +

v v v

1 2 4

2

11 13 14

21 23 24

31 33 34

v v v

v v v +

v v v1 3 4

1/22

12 13 14

22 23 24

32 33 34

v v v

v v v

v v v2 3 4

Efficient computation of Efficient computation of GALIGALIFor k deviation vectors:

ˆ ˆ ˆ )≤ ≤

∧ ∧ ∧ ⋅

∑

1 2 k

1 2 k

1 2 k

1 2 k

1/22

1 i 1 i 1 i

2 i 2 i 2 i T1 2 k

1 i <i < <i 2N

k i k i k i

v v v

v v vv v v = = det(A A

v v v⋯

⋯

⋯⋯

⋮ ⋮ ⋮

⋯

the ‘norm’ of the wedge product is given by:

ˆ ˆˆ

ˆ ˆˆ

ˆ ˆˆ

⋅ ⋅

11 12 1 2N 1 11

21 22 2 2N 2 22

k1 k2 k 2N 2N 2Nk

v v v e ev

v v v e ev = = A

v v v e ev

⋯

⋯

⋮ ⋮ ⋮ ⋮ ⋮⋮

⋯

Efficient computation of Efficient computation of GALIGALIFrom Singular Value Decomposition (SVD) of AT we get:

) ) )

) , , ) )

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ∏

T T T T T

k2 T 2 2 2 T 2

1 2 k ii=1

det(A A = det(V W U U W V = det(V W I W V =

det(V W V = det(V diag(w w w V = w…

where U is a column-orthogonal 2N×k matrix (UT·U=I), VT is a k×korthogonal matrix (V·VT=I), and W is a diagonal k×k matrix withpositive or zero elements, the so-calledsingular values.So, we get:

⋅ ⋅T TA = U W V

Thus, GALI k is computed by (Ch.S., Bountis, Antonopoulos, 2008,Eur. Phys. J. Sp. Top):

( ) ( )⋅ ⇒ =∑∏k k

Tk i k i

i=1i=1

GALI = det(A A ) = w log GALI log w

Behavior of GALIBehavior of GALI kk for for chaotic motionchaotic motion

GALI k (2≤k≤2N) tends exponentially to zero withexponents that involve the values of the first k largestLyapunov exponentsσ1, σ2, …, σk :

[ ]1 2 1 3 1 k- (σ -σ )+(σ -σ )+...+(σ -σ ) tkGALI (t) e∝∝∝∝

The above relation is valid even if some Lyapunovexponents are equal, or very close to each other.

Behavior of GALIBehavior of GALI kk for for chaotic motionchaotic motionN particles Fermi-Pasta-Ulam (FPU) system:

with fixed boundary conditions, N=8 and β=1.5.

( ) ( )

∑ ∑N N

2 42i i+1 i i+1 i

i=1 i=0

1 1 βH = p + q - q + q - q

2 2 4

Behavior of GALIBehavior of GALI kk forfor regular motionregular motionIf the motion occurs on an s-dimensional torus with s≤≤≤≤N then thebehavior of GALI k is given by (Ch.S., Bountis, Antonopoulos, 2008,Eur. Phys. J. Sp. Top.):

≤ ≤

≤

≤

k k-s

2(k -N)

constant if 2 k s

1GALI (t) if s < k 2N -s

t1

if 2N - s < k 2N t

∝∝∝∝

while in the common case with s=Nwe have :

≤ ≤

≤k

2(k -N)

constant if 2 k NGALI (t) 1

if N < k 2Nt

∝∝∝∝

Behavior of GALIBehavior of GALI kk for for regular motionregular motionN=8 FPU system: The unperturbed Hamiltonian (β=0) is written as asum of the so-calledharmonic energies Ei:

( )2 2 2i i i i

1E = P +ω Q , i = 1, ...,N

2with:

∑ ∑

N N

i k i k ik=1 k=1

2 kiπ 2 kiπ iπQ = q sin , P = p sin , ω = 2sin

N +1 N +1 N +1 N +1 2(N +1)

Global dynamicsGlobal dynamics• GALI 2 (practically equivalent to the use of SALI)

• GALI NChaotic motion: GALI NÆ0 (exponential decay)Regular motion: GALI NÆconstantπ0

3D HamiltonianSubspace q3=p3=0, p2≥0 for t=1000.

Global dynamicsGlobal dynamicsGALI k with k>N

The index tends to zero both forregular and chaotic orbits but withcompletely different time rates:

Chaotic motion: exponential decayRegular motion: power law

2D Hamiltonian (Hénon-Heiles)Time needed for GALI4<10-12

Behavior of GALIBehavior of GALI kk

Regular motion:GALI k →constant ≠ 0 or GALI k →0 power law decay

≤ ≤

≤

≤

k k -s

2(k -N)

constant if 2 k s

1GALI (t) if s < k 2N -s

t1

if 2N -s < k 2N t

∝∝∝∝

[ ]1 2 1 3 1 k- (σ -σ )+(σ -σ )+...+(σ -σ ) tkGALI (t) e∝∝∝∝

Chaotic motion:GALI k→0 exponential decay

Regular motionRegular motion on lowon low--dimensional dimensional toritori

A regular orbit lying on a 2-dimensional torus for the N=8 FPU system.

Regular motionRegular motion on lowon low--dimensional dimensional toritori

A regular orbit lying on a 4-dimensional torus for the N=8 FPU system.

Locating lowLocating low--dimensional toridimensional toriOrbits with q 1=q2=0.1, p1=p2=p3=0, H=0.010075 for the N=4 FPU system (Gerlach, Eggl, Ch.S., 2012, Int. J. Bif. Chaos).

SummarySummary• GALI k indices :

� candistinguish rapidly and with certainty between regular and chaotic motion

� can be used to characterizeindividual orbits as well as"chart" chaotic andregular domains in phase space.

� are perfectly suited for studying the global dynamics of multidimentonalsystems

� can identify regular motion on low–dimensional tori

• SALI/GALI methods have been successfullyapplied to a variety of conservative dynamicalsystemsof

� Celestial Mechanics (e.g. Széll et al., 2004, MNRAS - Souliset al., 2008, Cel. Mech. Dyn.Astr. - Libert et al., 2011, MNRAS)

� Galactic Dynamics (e.g. Capuzzo-Dolcetta et al., 2007, Astroph. J. - Carpintero, 2008,MNRAS - Manos & Athanassoula, 2011, MNRAS)

� Nuclear Physics (e.g. Macek et al., 2007, Phys. Rev. C - Stránský et al., 2007, Phys.Atom. Nucl. - Stránský et al., 2009, Phys. Rev. E)

� Statistical Physics (e.g. Manos & Ruffo, 2010, Trans. Theory Stat. Phys.)

Main referencesMain references• Bountis T.C. & Ch.S. (2012) ‘Complex Hamiltonian Dynamics’, Chapter

5, Springer Series in Synergetics

• SALI� Ch.S. (2001) J. Phys. A, 34, 10029� Ch.S., Antonopoulos Ch., Bountis T. C. & Vrahatis M. N. (2003) Prog.

Theor. Phys. Supp., 150, 439� Ch.S., Antonopoulos Ch., Bountis T. C. & Vrahatis M. N. (2004) J.

Phys. A, 37, 6269� Bountis T. & Ch.S. (2006) Nucl. Inst Meth. Phys Res. A, 561, 173� Boreaux J., Carletti T., Ch.S. &Vittot M. (2012) Com. Nonlin. Sci.

Num. Sim., 17, 1725� Boreaux J., Carletti T., Ch.S., Papaphilippou Y. & Vittot M. (2012)

Int. J. Bif. Chaos, 22, 1250219

• GALI� Ch.S., Bountis T. C. & Antonopoulos Ch. (2007) Physica D, 231, 30-54� Ch.S., Bountis T. C. & Antonopoulos Ch. (2008) Eur. Phys. J. Sp.

Top., 165, 5-14� Gerlach E., Eggl S. & Ch.S. (2012) Int. J. Bif. Chaos, 22, 1250216� Manos T., Ch.S. & Antonopoulos Ch. (2012) Int. J. Bif. Chaos,22,

1250218[Behavior of GALI for periodic orbits]� Manos T., Bountis T. & Ch.S. (2013) J. Phys. A, 46, 254017[Behavior

of GALI for time dependent Hamiltonians]-> Talk of T. Manos onThursday 20 June

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